Download Algebra for the Practical Man by James Edgar Thompson PDF

By James Edgar Thompson

This publication has been designed to give many of the branches of primary arithmetic so they should be comprehensible with no the help of a teacher.
the writer, in his adventure as an engineer and as a instructor of arithmetic to pupil and laymen in night faculties and in schools, has heard a lot of those males convey to hope for books that could be used for domestic learn, and the sequence of books to which this quantity belongs is the result of an try and meet their wishes and wishes.
Данная книга была спроектирована с целью подачи различных ветвей фундаментальной математики, так чтобы их можно было понять без помощи учителя.
Автор имея опыт инженера и учителя математики для учеников различного уровня подготовки в вечерних школах и коледжах, учел просьбы по проэктированию этой книги таким образом чтобы она могла быть использованна для домашнего самообучения, и серия книг включая и эту результат многочисленных попыток удовлетворить эти просьбы.

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Example text

Theorem 14. I A G be a group, K a normal subgroup of G and L any subgroyp of G; let x be the natural homomorphism of G onto GJK. Then 4. GROUPS OPERATING ON A SET 35 the subgroup &lx(L))of G is the set of all products st, for s E L, t~ K; it is also the set of all products ts, for t E K , s E L. If xL is the restriction of x to L, then x L is an epimorphism of L on x(L) whose kernel is K n L. A necessary -1 and sufficient condition for K to be Contained in L is that L = x(x(L));the mapping L + x(L) establishes a bijection of the set of all subgroups of G containing K on the set of all subgroups of G/K.

Let be a commutative monoid in which the cancellation law holds. 1, let y(a, b) be the set of elements (a’, b’) E M x M such that ab’ = a‘b. (a, b), we have y(a‘, b’) = y(a, 6). For, let (a’’, b”) be in -;(a’, b’); then a‘b“ = d ’ b ’ , whence barb’‘= bd‘b’; but we have ab’ = arb by assumption, so that b’ab‘‘ = bd’b’, and therefore ab“ = a”b by the cancellation law, which shows that y(a‘, b‘) c y(a, b). (a‘, b’) and -:(a, b) = -;(a’, b’). Thus, if the sets y(a, b) and y(a,, b,) have a n element (a’, b’) in common, these sets are equal, which shows th a t the sets y(a, b), for all (a, b) E M x M, are mutually disjoint.

We use the term subgroup for a group which may be defined in this manner. EXAMPLES : a) Let A be a monoid. Then the set of automorphisms of A is a subgroup of the set of all permutations of A . b) Let be the set of integers, and let k be any element of Then the set of multiples of k, i. e. the set of all elements pk, for all p E is a sub- z z. z, 28 11. GROUPS z, + + group of Z. k = 0. c ) Let S be a set, and S’ any subset of S. Then the set of permutations w of S such that w(S’) = S’ is a subgroup P’ of the group P of permutations of S.

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